Abstract

For a laser that fires a short pulse at time 0 into a homogeneous cloud with specified scattering and absorption parameters, this paper addresses the problem of theoretically calculating Jn(t), the nth-order backscattered power measured at any time t > 0. The backscattered power is assumed to be measured by a small receiver, which is colocated with the laser and which is fitted with a forward-looking conical baffle of adjustable opening angle. The approach taken here to calculate Jn(t) is somewhat unusual in that it is not based on the radiation-transfer equation but rather on the premise that the laser pulse consists of propagating photons, which are scattered and absorbed in a probabilistic manner by the cloud particles. Polarization effects have not been considered. By using straightforward physical arguments together with rigorous analytical techniques from the theory of random variables, an exact formula is derived for Jn(t). For n ≥ 2 this formula is a well-behaved (3n − 4)-dimensional integral. The computational feasibility of this integral formula is demonstrated by using it to evaluate Jn(t)/J1(t) for a model cloud of isotropically scattering particles; for that case an analytical formula is obtained for n = 2, and a Monte Carlo integration program is employed to obtain numerical results for n = 3, …, 6.

© 1985 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  2. K. N. Liou, R. M. Schotland, “Multiple backscattering and depolarization from water clouds for a pulsed lidar system,” J. Atmos. Sci. 28, 772–784 (1971).
    [Crossref]
  3. E. W. Eloranta, “Calculation of doubly scattered lidar returns,” Ph.D. dissertation (University of Wisconsin, Madison, Wisconsin, 1972).
  4. J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmospheres,” J. Atmos. Sci. 33, 1763–1771 (1976).
    [Crossref]
  5. L. L. Carter, H. G. Horak, M. T. Sandford, “An adjoint Monte Carlo treatment of the equations of radiative transfer for polarized light,” J. Comput. Phys. 26, 119–138 (1978).
    [Crossref]
  6. Q. Cai, K. N. Liou, “Theory of time-dependent multiple backscattering from clouds,” J. Atmos. Sci. 38, 1452–1466 (1981).
    [Crossref]
  7. G. N. Plass, G. W. Kattawar, “Reflection of light pulses from clouds,” Appl. Opt. 10, 2304–2310 (1971).
    [Crossref] [PubMed]
  8. W. G. Blättner, D. G. Collins, M. B. Wells, “The effects of multiple scattering on backscatter lidar measurements in fog,” Proj. Rep. RRA-T7402 (Radiation Research Associates, 1974),available as AD-78180116GI (National Technical Information Service, Springfield, Va.).
  9. K. E. Kunkle, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
    [Crossref]
  10. D. T. Gillespie, “A theorem for physicists in the theory of random variables,” Am. J. Phys. 51, 520–533 (1983).
    [Crossref]
  11. D. T. Gillespie, “Analytic reduction of the monostatic lidar multiple backscattering integral,” Rep. NWC TP 6605 (Naval Weapons Center, China Lake, Calif., 1985).
  12. As used here, the term “scattering” refers only to elastic encounters between a photon and a cloud particle in which the photon’s energy and hence wavelength do not change. An inelastic encounter is considered to be an “absorption,” since any shift in a photon’s wavelength is presumed to render it invisible to the monochromatic receiver.
  13. Letting P(u) denote the probability on the right-hand side of Eq. (4), then the multiplication law of probability theory implies that P(u+ du) is equal to P(u) times the probability (1 − β du) that nothing will happen to the photon between u and u+ du. This equality leads immediately to the differential equation dP/du= − βP, whose solution, subject to the obvious requirement P(0) = 1, is exp(−βu).
  14. These variables are algebraically independent because they satisfy no algebraic equation that can be solved for some one of the variables in terms of the others. However, these variables are not statistically independent, since their joint probability density function (∝ Qn) cannot be factored into a product of single-variable functions.
  15. We should really distinguish between a random variable and its possible values by using different symbols for each. For example, T might be used to represent the random variable “total photon travel time,” whereas t could be the dummy real variable representing the possible values that T may assume. However, to avoid a proliferation of notation, we shall use the same symbol t for both of these quantitites, hoping that our meaning will always be clear from the context.
  16. The rule for transforming delta functions is discussed in D. T. Gillespie, “Addenda to ‘A theorem for physicists in the theory of random variables,’ ” Rep. NWC TP 6462 (Naval Weapons Center, China Lake, Calif., 1983).
  17. D. T. Gillespie, “The Monte Carlo method of evaluating integrals,” Rep. NWC TP 5714 (Naval Weapons Center, China Lake, Calif., 1975).Available from National Technical Information Center, Springfield, Va.

1983 (1)

D. T. Gillespie, “A theorem for physicists in the theory of random variables,” Am. J. Phys. 51, 520–533 (1983).
[Crossref]

1981 (1)

Q. Cai, K. N. Liou, “Theory of time-dependent multiple backscattering from clouds,” J. Atmos. Sci. 38, 1452–1466 (1981).
[Crossref]

1978 (1)

L. L. Carter, H. G. Horak, M. T. Sandford, “An adjoint Monte Carlo treatment of the equations of radiative transfer for polarized light,” J. Comput. Phys. 26, 119–138 (1978).
[Crossref]

1976 (2)

K. E. Kunkle, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[Crossref]

J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmospheres,” J. Atmos. Sci. 33, 1763–1771 (1976).
[Crossref]

1971 (2)

K. N. Liou, R. M. Schotland, “Multiple backscattering and depolarization from water clouds for a pulsed lidar system,” J. Atmos. Sci. 28, 772–784 (1971).
[Crossref]

G. N. Plass, G. W. Kattawar, “Reflection of light pulses from clouds,” Appl. Opt. 10, 2304–2310 (1971).
[Crossref] [PubMed]

Blättner, W. G.

W. G. Blättner, D. G. Collins, M. B. Wells, “The effects of multiple scattering on backscatter lidar measurements in fog,” Proj. Rep. RRA-T7402 (Radiation Research Associates, 1974),available as AD-78180116GI (National Technical Information Service, Springfield, Va.).

Cai, Q.

Q. Cai, K. N. Liou, “Theory of time-dependent multiple backscattering from clouds,” J. Atmos. Sci. 38, 1452–1466 (1981).
[Crossref]

Carter, L. L.

L. L. Carter, H. G. Horak, M. T. Sandford, “An adjoint Monte Carlo treatment of the equations of radiative transfer for polarized light,” J. Comput. Phys. 26, 119–138 (1978).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Collins, D. G.

W. G. Blättner, D. G. Collins, M. B. Wells, “The effects of multiple scattering on backscatter lidar measurements in fog,” Proj. Rep. RRA-T7402 (Radiation Research Associates, 1974),available as AD-78180116GI (National Technical Information Service, Springfield, Va.).

Eloranta, E. W.

E. W. Eloranta, “Calculation of doubly scattered lidar returns,” Ph.D. dissertation (University of Wisconsin, Madison, Wisconsin, 1972).

Gillespie, D. T.

D. T. Gillespie, “A theorem for physicists in the theory of random variables,” Am. J. Phys. 51, 520–533 (1983).
[Crossref]

D. T. Gillespie, “Analytic reduction of the monostatic lidar multiple backscattering integral,” Rep. NWC TP 6605 (Naval Weapons Center, China Lake, Calif., 1985).

The rule for transforming delta functions is discussed in D. T. Gillespie, “Addenda to ‘A theorem for physicists in the theory of random variables,’ ” Rep. NWC TP 6462 (Naval Weapons Center, China Lake, Calif., 1983).

D. T. Gillespie, “The Monte Carlo method of evaluating integrals,” Rep. NWC TP 5714 (Naval Weapons Center, China Lake, Calif., 1975).Available from National Technical Information Center, Springfield, Va.

Horak, H. G.

L. L. Carter, H. G. Horak, M. T. Sandford, “An adjoint Monte Carlo treatment of the equations of radiative transfer for polarized light,” J. Comput. Phys. 26, 119–138 (1978).
[Crossref]

Kattawar, G. W.

Kunkle, K. E.

K. E. Kunkle, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[Crossref]

Liou, K. N.

Q. Cai, K. N. Liou, “Theory of time-dependent multiple backscattering from clouds,” J. Atmos. Sci. 38, 1452–1466 (1981).
[Crossref]

K. N. Liou, R. M. Schotland, “Multiple backscattering and depolarization from water clouds for a pulsed lidar system,” J. Atmos. Sci. 28, 772–784 (1971).
[Crossref]

Plass, G. N.

Sandford, M. T.

L. L. Carter, H. G. Horak, M. T. Sandford, “An adjoint Monte Carlo treatment of the equations of radiative transfer for polarized light,” J. Comput. Phys. 26, 119–138 (1978).
[Crossref]

Schotland, R. M.

K. N. Liou, R. M. Schotland, “Multiple backscattering and depolarization from water clouds for a pulsed lidar system,” J. Atmos. Sci. 28, 772–784 (1971).
[Crossref]

Weinman, J. A.

J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmospheres,” J. Atmos. Sci. 33, 1763–1771 (1976).
[Crossref]

K. E. Kunkle, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[Crossref]

Wells, M. B.

W. G. Blättner, D. G. Collins, M. B. Wells, “The effects of multiple scattering on backscatter lidar measurements in fog,” Proj. Rep. RRA-T7402 (Radiation Research Associates, 1974),available as AD-78180116GI (National Technical Information Service, Springfield, Va.).

Am. J. Phys. (1)

D. T. Gillespie, “A theorem for physicists in the theory of random variables,” Am. J. Phys. 51, 520–533 (1983).
[Crossref]

Appl. Opt. (1)

J. Atmos. Sci. (4)

Q. Cai, K. N. Liou, “Theory of time-dependent multiple backscattering from clouds,” J. Atmos. Sci. 38, 1452–1466 (1981).
[Crossref]

K. N. Liou, R. M. Schotland, “Multiple backscattering and depolarization from water clouds for a pulsed lidar system,” J. Atmos. Sci. 28, 772–784 (1971).
[Crossref]

J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmospheres,” J. Atmos. Sci. 33, 1763–1771 (1976).
[Crossref]

K. E. Kunkle, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[Crossref]

J. Comput. Phys. (1)

L. L. Carter, H. G. Horak, M. T. Sandford, “An adjoint Monte Carlo treatment of the equations of radiative transfer for polarized light,” J. Comput. Phys. 26, 119–138 (1978).
[Crossref]

Other (10)

E. W. Eloranta, “Calculation of doubly scattered lidar returns,” Ph.D. dissertation (University of Wisconsin, Madison, Wisconsin, 1972).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

W. G. Blättner, D. G. Collins, M. B. Wells, “The effects of multiple scattering on backscatter lidar measurements in fog,” Proj. Rep. RRA-T7402 (Radiation Research Associates, 1974),available as AD-78180116GI (National Technical Information Service, Springfield, Va.).

D. T. Gillespie, “Analytic reduction of the monostatic lidar multiple backscattering integral,” Rep. NWC TP 6605 (Naval Weapons Center, China Lake, Calif., 1985).

As used here, the term “scattering” refers only to elastic encounters between a photon and a cloud particle in which the photon’s energy and hence wavelength do not change. An inelastic encounter is considered to be an “absorption,” since any shift in a photon’s wavelength is presumed to render it invisible to the monochromatic receiver.

Letting P(u) denote the probability on the right-hand side of Eq. (4), then the multiplication law of probability theory implies that P(u+ du) is equal to P(u) times the probability (1 − β du) that nothing will happen to the photon between u and u+ du. This equality leads immediately to the differential equation dP/du= − βP, whose solution, subject to the obvious requirement P(0) = 1, is exp(−βu).

These variables are algebraically independent because they satisfy no algebraic equation that can be solved for some one of the variables in terms of the others. However, these variables are not statistically independent, since their joint probability density function (∝ Qn) cannot be factored into a product of single-variable functions.

We should really distinguish between a random variable and its possible values by using different symbols for each. For example, T might be used to represent the random variable “total photon travel time,” whereas t could be the dummy real variable representing the possible values that T may assume. However, to avoid a proliferation of notation, we shall use the same symbol t for both of these quantitites, hoping that our meaning will always be clear from the context.

The rule for transforming delta functions is discussed in D. T. Gillespie, “Addenda to ‘A theorem for physicists in the theory of random variables,’ ” Rep. NWC TP 6462 (Naval Weapons Center, China Lake, Calif., 1983).

D. T. Gillespie, “The Monte Carlo method of evaluating integrals,” Rep. NWC TP 5714 (Naval Weapons Center, China Lake, Calif., 1975).Available from National Technical Information Center, Springfield, Va.

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Figures (8)

Fig. 1
Fig. 1

The lidar configuration considered in this paper. The situation in which the ground is not present and the cloud fills all xyz space can be treated as a special case of this configuration.

Fig. 2
Fig. 2

Trajectory of a photon that scatters exactly once in the cloud and then returns to the ground plane infinitesimally close to the origin.

Fig. 3
Fig. 3

Trajectory of a photon that scatters exactly n times in the cloud and then returns to the ground plane at the point S = (x, y, 0). The ith scattering occurs at point Si, and the vector from Si to Si+1 is denoted by uiêiui, where êi is a vector of unit length.

Fig. 4
Fig. 4

Geometric interpretation of the relations among the principal variables in Eqs. (45) and (46) for (a) n = 2, (b) n = 3, and (c) n = 4. The photon’s journey begins and ends at point O, and the ith scattering occurs at point Si (i = 1, …, n). The vector vi = SiSi+1 has magnitude υi and unit direction êi. All lengths have been scaled dimensionless, with OSn = C0 having length 1. The direction of êi for i ≥ 1 is measured by angles (θi, ϕi) relative to the polar axis ei−1 and by angles (θi′, ϕi) relative to the polar axis CiSiSn. Not shown (for reasons of graphical clarity) are the vectors BiOSi (i = 1, …, n − 1) and the angles νi (i = 1, …, n − 2); νi is the angle between Ci+1 and the line perpendicular to vi from Sn. The main coordinate frame is defined so that ê0 points along the z axis and C0 lies in the xz plane. The quantity V defined in Eq. (45l) is the circumference of the (generally nonplanar) figure OS1 … SnO.

Fig. 5
Fig. 5

A plot of log10K(n, ψ0) versus log10ψ0 for n = 2, 3, …, 6 for an isotropically scattering ground cloud. The plotted points are from the data in Table 1, which in turn were obtained from Eq. (60) for n = 2 and from Monte Carlo evaluations of Eq. (57) for n = 3, …, 6. The solid curves are smooth interpolations through the data points. The dashed lines over the n = 2, 4, 5, 6 curves are plots of the estimated small-angle asymptotes in Eqs. (65); the n = 3 curve does not appear to have a small-angle constant-slope asymptote, evincing instead the logarithmic behavior of expression (67).

Fig. 6
Fig. 6

A plot of log10K(n, ψ0) versus log10ψ0 for n = 2, 3, …, 6 for an isotropically scattering enveloping cloud. The plotted points are from the data in Table 2, which in turn were obtained from Eq. (60) for n = 2 and from Monte Carlo evaluations of Eq. (57), without the I functions, for n = 3, …, 6. The full curves are smooth interpolations through the data points. The dashed lines over the n = 2, 4, 5, 6 curves are plots of the estimated small-angle asymptotes in expressions (66); the n = 3 curve does not appear to have a small-angle constant-slope asymptote, evincing instead the logarithmic behavior of expression (67).

Fig. 7
Fig. 7

A log-log scale plot of K(n, ψ0) versus ψ0 for n = 2, 3, …, 6 for an isotropically scattering ground cloud (lower curve for each n) and an isotropically scattering enveloping cloud (upper curve for each n). This plot is essentially a superposition of portions of Figs. 5 and 6 for which ψ0 ≥ 1°.

Fig. 8
Fig. 8

Log-log scale plots of Jn*(z*)/J1*(z*) versus z* ≡ βsct/2 for n = 2, 3, …, 6 for an isotropically scattering ground cloud with (a) ψ0 = 0.001 rad, (b) ψ0 = 0.01 rad, (c) ψ0 = 0.1 rad, and (d) ψ0 = π/2 rad. Physically, Jn*(z*) is the n-scattered radiation power measured at the receiver at the moment when the scattering altitude of co-arriving singly scattered radiation equals z* × βs−1. The curves are plots of Eqs. (70), using the K(n, ψ0) values from Table 1.

Tables (2)

Tables Icon

Table 1 Calculated Values of K(n, ψ0) for an Isotropically Scattering Ground Clouda,b

Tables Icon

Table 2 Calculated Values of K(n, ψ0) for an Isotropically Scattering Enveloping Clouda,b

Equations (120)

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J n ( t ) the power , measured by the receiver at time t , that has been scattered exactly n times by the could particles ( t > 0 ; n = 1 , 2 , ) .
δ ( x x 0 ) = 0 if x x 0 ,
f ( x ) δ ( x x 0 ) d x f ( x 0 )
I ( inequality ) { 1 if inequality is satisfied 0 if inequality is not satisfied .
exp ( β u ) = probability that a photon will move a distance u in the cloud without being either scattered or absorbed ,
β s d u × f ( θ ) sin θ d θ d ϕ = probability that a photon will be scattered , in the next infinitesimal distance d u , into the solid angle sin θ d θ d ϕ in the polar direction ( θ , ϕ ) relative to its present direction of travel . ( 0 θ π ; 0 ϕ < 2 π ) .
( ρ k V c ) × ( σ s , k d u / V c ) = ρ k σ s , k d u .
ρ k σ s , k d u = probability that the photon will be scattered in d u by a type - k particle .
ρ k σ a , k d u = probability that the photon will be absorbed in d u by a type - k particle .
β s k ρ k σ s , k ,
β a k ρ k σ a , k ,
β β s + β a k ρ k ( σ s , k + σ a , k ) ,
β s d u = probability that the photon will be scattered in the next d u ,
β a d u = probability that the photon will be absorbed in the next d u ,
β d u = probability that the photon will be either scattered or absorbed in the next d u .
f k ( θ ) sin θ d θ d ϕ probability that a photon , which has just been scattered by a type - k particle , will have its new direction of travel pointing in the infinitesimal solid angle sin θ d θ d ϕ , at polar angle θ and azimuthal angle ϕ relative to its previous direction of travel . ( 0 θ π ; 0 ϕ < 2 π ) .
0 π f k ( θ ) sin θ d θ = ( 2 π ) 1 .
ρ k σ s , k d u f k ( θ ) sin θ d θ d ϕ = probability that a photon will be scattered in the next d u by a type - k particle in to the solid angle sin θ d θ d ϕ in the polar direction ( θ , ϕ ) relative to its present direction of travel ( 0 θ π ; 0 ϕ < 2 π ) .
k { ρ k σ s , k d u × f k ( θ ) sin θ d θ d ϕ } β s d u { k ( ρ s σ s , k / β s ) f k ( θ ) } sin θ d θ d ϕ .
f ( θ ) k ( ρ k σ s , k / β s ) f k ( θ ) ,
0 π f ( θ ) sin θ d θ = ( 2 π ) 1 ,
β s = k ρ k σ s , k ,
β = k ρ k ( σ s , k + σ a , k ) ,
f ( θ ) = β s 1 k ρ k σ s , k f k ( θ ) .
β s = 0 σ s ( r ) ρ ( r ) d r ,
β = 0 [ σ s ( r ) + σ a ( r ) ] ρ ( r ) d r ,
f ( θ ) = β s 1 0 f ( θ ; r ) σ s ( r ) ρ ( r ) d r .
N 0 p ( τ ) d τ = average number of photons emitted by the laser in the infinitesimal time interval ( τ , τ + d τ ) .
0 Δ p ( t ) d t = 1 ,
P n ( t , x , y ) d t d x d y probability that a photon , which is emitted by the laser a time 0 , will suffer exactly n scattering sin the cloud and then arrive at the ground in the infinitesimal time interval ( t , t + d t ) , and in the infinitesimal area element d x d y at point ( x , y ) , and at anangle less than ψ 0 with the verticle .
disk P n ( t τ , x , y ) d x d y R n ( t τ ) ,
R n ( t τ ) d t = probability that a photon , which is emittetd by the laser at time τ , will suffer exactly n scatterings in the cloud and then be detected at the receiver in the infinitesimal time interval ( t , t + d t ) = average fraction of the N 0 p ( τ ) d τ photons emitted by the laser in ( τ , τ + d τ ) that will suffer exactly n scattering sin the cloud and then be detected at the receiver in ( t , t + d t ) .
τ = 0 Δ N 0 p ( τ ) d τ × R n ( t τ ) d t = N 0 d t 0 Δ d τ p ( τ ) R n ( t τ ) .
J n ( t ) = N 0 0 Δ d τ p ( τ ) R n ( t τ ) ( t > Δ ) ,
J n ( t ) = N 0 0 Δ d τ p ( τ ) disk P n ( t τ , x , y ) d x d y P n ( t τ , x , y ) d x d y ( t > Δ ) .
disk P n ( t τ , x , y ) d x d y π r 0 2 × P n ( t τ , 0 , 0 ) .
t r 0 / c + Δ ,
J n ( t ) = ( π r 0 2 ) N 0 0 Δ d τ p ( τ ) P n ( t τ , 0 , 0 ) [ t r 0 / c + Δ ] .
J n ( t ) = ( π r 0 2 ) N 0 P n ( t , 0 , 0 ) [ t r 0 / c , Δ 0 ] .
J n ( t ) / J 1 ( t ) = P n ( t , 0 , 0 ) / P 1 ( t , 0 , 0 ) [ t r 0 / c , Δ 0 ] .
P 1 ( t , 0 , 0 ) d t d x d y probability that a photon , emitted by the laser at time 0 , will scatter exactly once in the cloud and then arrive , between times t and t + d t , in the infinitesimal area element d x d y centered on the origin .
z = c t / 2 and d z = c d t / 2 .
d 2 Ω = ( d x d y ) / z 2 .
P 1 ( t , 0 , 0 ) d t d x d y = exp [ β ( z b ) ] × β s d z × f ( π ) d 2 Ω × exp [ β ( z b ) ] × I ( z > b ) .
P 1 ( t , 0 , 0 ) = I ( c t > 2 b ) ( c β s / 2 ) ( 2 / c t ) 2 × exp [ β ( c t 2 b ) ] f ( π ) .
J 1 ( t ) = I ( c t > 2 b ) ( π r 0 2 ) N 0 2 c β s ( c t ) 2 × exp [ β ( c t 2 b ) ] f ( π ) [ t r 0 / c , Δ 0 ] .
u i S i S i + 1 u i ê i ( i = 0 , 1 , , n ) .
ê 0 = .
x ̂ 1 = x ̂ , ŷ 1 = ŷ , 1 = ,
i = ê i 1 ŷ i = ( × ê i 1 ) / | × ê i 1 | x ̂ i = ŷ i × i } . ( i = 2 , , n ) .
ê i = x ̂ i sin θ i cos ϕ i + ŷ i sin θ i sin ϕ i + i cos θ i ( i = 1 , , n ) .
cos ψ = ê n ( 0 ψ < π / 2 ) .
0 = OS = i = 0 n u i = i = 0 n u i ( ê i ) .
u n = ( ê n ) 1 i = 0 n 1 u i ( ê i ) .
Q n ( u 0 , θ 1 , ϕ 1 , , u n 1 , θ n , ϕ n ) d u 0 d θ 1 d ϕ 1 d u n 1 d θ n d ϕ n probability that a photon , which is emitted from the laser at time 0 , will scatter exactly n times in the cloud in such a way that , for each i = 1 , , n the free path length of the photon just before the i th scattering is between u i 1 and u i 1 + d u i 1 and its heading just after the i th scattering is in the polar solid angle sin θ i d θ i d ϕ i at ( θ i , ϕ i ) , after which the photon returns freely to the ground at an angle less than ψ 0 with the vertical .
Q n ( u 0 , θ 1 , ϕ 1 , , u n 1 , θ n , ϕ n ) d u 0 d θ 1 d ϕ 1 d u n 1 d θ n d ϕ n = exp ( β b ) × exp [ β ( u n b sec ψ ) ] × i = 1 n [ exp ( β u i 1 ) × I ( j = 0 i 1 u j > b ) × β s d u i 1 × f ( θ i ) sin θ i d θ i d ϕ i × I ( u i 1 0 ) I ( 0 θ i π ) I ( 0 ϕ i < 2 π ) ] × I ( cos ψ > cos ψ 0 ) .
Q n ( u 0 , θ 1 , ϕ 1 , , u n 1 , θ n , ϕ n ) = β s n exp ( β i = 0 n u i ) exp { β b [ 1 ( ê n ) 1 ] } i = 1 n [ f ( θ i ) sin θ i ] × I ( ê n > cos ψ 0 ) i = 1 n [ I ( j = 0 i 1 u j ( ê j ) > b ) × I ( u i 1 0 ) I ( 0 θ i π ) I ( 0 ϕ i < 2 π ) ] .
Y i = f i ( X 1 , , X n ) ( i = 1 , , m ) ,
P ( y 1 , , y m ) = d x 1 d x n Q ( x 1 , , x n ) × i = 1 m δ [ y i f i ( x 1 , , x n ) ] .
K n probability that a photon , which is emitted from the laser at time 0 , will suffer exactly n scatterings in the cloud and then return freely to the ground at an angle less than ψ 0 with the vertical .
i = 0 n u i = c t ; x ̂ i = 0 n û i = x ; ŷ i = 0 n u i = y .
t = c 1 i = 0 n u i ; x = i = 0 n u i ( x ̂ ê i ) ; y = i = 0 n u i ( ŷ ê i ) .
K n 1 P n ( t , x , y ) = d u 0 d ϕ n × K n 1 Q n ( u 0 , , ϕ n ) δ ( t c 1 i = 0 n u i ) × δ ( x i = 0 n u i ( x ê i ) ) δ ( y i = 0 n u i ( y ê i ) ) .
P n ( t , x , y ) = β s n 0 d u 0 0 d u n 1 0 π d θ 1 × 0 2 π d θ 1 0 π d θ n 0 2 π d ϕ n exp ( β i = 0 n u i ) × exp { β b [ 1 ( ê n ) 1 ] } i = 1 n { f ( θ i ) sin θ i × I ( j = 0 i 1 u j ( ê j ) > b ) } I ( ê n > cos ψ 0 ) × δ ( t c 1 i = 0 n u i ) δ ( x i = 0 n u i ( x ̂ ê i ) ) × δ ( y i = 0 n u i ( ŷ ê i ) ) ,
P n ( t , x , y ) = β s n exp ( β c t ) 0 d u 0 0 d u n 1 × 0 π d θ 1 0 2 π d ϕ 1 0 π d ϕ n 0 2 π d ϕ n × exp { β b [ 1 ( ê n ) 1 ] } i = 1 n { f ( θ i ) sin θ i × I ( j = 0 i 1 u j ( ê j ) > b ) } I ( ê n > cos ψ 0 ) × δ ( t c 1 i = 0 n u i [ 1 ( ê i ) ( ê n ) 1 ] ) × δ ( x i = 0 n 1 u i [ ( x ̂ ê i ) ( ê i ) ( x ̂ ê n ) ( ê n ) 1 ] ) × δ ( y i = 0 n 1 u i [ ( ŷ ê i ) ( ê i ) ( ŷ ê n ) ( ê n ) 1 ] ) .
υ 0 , , υ n 2 , θ 1 , ϕ 1 , θ n 2 , ϕ n 2 , ψ .
ν 0 , , ν n 2 , θ 1 , ϕ 1 , , θ n 2 , ϕ n 2 , θ 0 ψ ,
P n ( t , 0 , 0 ) = 2 π β s n c ( c t ) n 3 exp ( β c t ) 0 ψ 0 d θ 0 × θ 0 π / 2 π / 2 d ν 0 { i = 1 n 2 0 π d θ i θ i π / 2 π / 2 d ν i 0 2 π d ϕ i } × ( i = 1 n 1 I ( B i , z > Vb / ct ) I ( cos θ 0 > Vb / ct ) × exp [ β b ( 1 + sec θ 0 ) ] × ( i = 1 n f ( θ i ) ) cos θ 0 { i = 1 n 2 C i V } ( n 2 ) ,
{ C 0 = x ̂ sin θ 0 + cos θ 0 , υ 0 = sin θ 0 tan ν 0 + cos θ 0 , ê 0 = ,
{ C i = C i 1 υ i 1 ê i 1 , υ i = C i sin θ i tan ν i + C i cos θ i , ê 1 = x ̂ e i , x + ŷ e i , y + e i , z [ see Eqs . ( 46 ) ] ( n 3 ; i = 1 , n 2 ) ,
{ C n 1 = C n 2 υ n 2 ê n 2 , υ n 1 = C n 1 , , e n 1 = C n 1 / C n 1 ,
B 1 = υ 0 ê 0 ,
B i = B i 1 + υ i 1 ê i 1 ( n 3 ; i = 1 , , n 1 ) ,
V = 1 + i = 0 n 1 υ i ,
θ i = arccos ( ê i 1 ê i ) ( i = 1 , , n 1 ) ,
θ n = arccos ( ê n 1 C 0 ) .
[ e i , x e i , y e i , z ] = [ c i , z c i , x / c i , x y c i , y / c i , x y c i , x c i , z c i , y / c i , x y c i , x / c i , x y c i , y c i , x y 0 c i , z ] [ sin θ i cos ϕ i sin θ i sin ϕ i cos θ i ] ( i = 1 , , n 2 ) ,
c i , x C i , x / C i , c i , y C i , y / C i , c i , z C i , z / C i , c i , x y ( C i , x 2 + C i , y 2 ) 1 / 2 / C i ,
κ 0 ψ 0 ( 2 π ψ 0 ) / π 2 ,
θ 0 = π [ 1 ( 1 κ 0 p 0 ) 1 / 2 ] ,
ν 0 = π [ 1 / 2 q 0 ( 1 κ 0 p 0 ) 1 / 2 ] .
θ i = π ( 1 p i 1 / 2 ) ,
ν i = π ( 1 / 2 q i p i 1 / 2 ) ,
ϕ i = 2 π w i
| ( θ 0 , ν 0 , θ 1 , ν 1 , ϕ 1 , , θ n 2 , ν n 2 , ϕ n 2 ) ( p 0 , q 0 , p 1 , q 1 , w 1 , , p n 2 , q n 2 , w n 2 ) | = π 3 n 4 κ 0 2 .
P n ( t , 0 , 0 ) = π 3 ( n 1 ) κ 0 β s n c ( c t ) n 3 exp ( β c t ) × 0 1 d p 0 0 1 d q 0 { i = 1 n 2 0 1 d p 1 0 1 d q i 0 1 d w i } × [ i = 1 n 1 I ( B i , z > V b / c t ) ] I ( cos θ 0 > V b / c t ) × exp [ β b ( 1 + sec θ 0 ) ] [ i = 1 n f ( θ i ) ] cos θ 0 { i = 1 n 2 C i V } ( n 2 ) .
c t / b > V ( cos θ 0 ) 1 V 2 ,
J 2 ( t ) J 1 ( t ) = π 3 κ 0 2 f ( π ) ( β s c t ) 0 1 d p 0 0 1 d p 0 I ( B 1 , z > V b / c t ) × I ( cos θ 0 > V b / c t ) exp [ β b ( sec θ 0 1 ) ] f ( θ 1 ) f ( θ 2 ) cos θ 0 ( t r 0 / c , Δ 0 )
J n ( t ) J n ( t ) = π 3 ( n 1 ) κ 0 2 f ( π ) ( β s c t ) n 1 0 1 d p 0 0 1 d q 0 × { i = 1 n 2 0 1 d p 0 0 1 d q i 0 1 d w i } × [ i = 1 n 1 I ( B i , z > V b / c t ) ] I ( cos θ 0 > V b / c t ) × exp [ β b ( sec θ 0 1 ) ] [ i = 1 n f ( θ i ) ] cos θ 0 [ i = 1 n 2 C i V ] ( n 3 ; t r 0 / c , Δ 0 ) .
For any given n 2 , J n ( t ) / J 1 ( t ) increases with t like t n 1 , if b = 0 or if the cloud completely surrounds the lidar , and increases somewhat faster than this if b > 0 .
f ( θ ) = ( 4 π ) 1 , b = 0 ,
B 1 , z = υ 0 > 0 , cos θ 0 > 0 ,
J n ( t ) / J 1 ( t ) = K ( n , ψ 0 ) ( β s c t ) n 1 ,
K ( 2 , ψ 0 ) π 2 κ 0 8 0 1 d p 0 0 1 d q 0 cos θ 0
K ( n , ψ 0 ) ( π 2 4 ) n 1 κ 0 2 0 1 d p 0 0 1 d q 0 × { i = 1 n 2 0 1 d p i 0 1 d q i 0 1 d w i } × cos θ 0 [ i = 1 n 2 I ( B i + 1 , z > 0 ) C i ] V ( n 2 ) ( n 3 ) .
J 1 ( t ) = N 0 ( π r 0 2 / 2 π ) c β s 3 ( β s c t ) 2 exp ( β c t ) ,
d θ 0 d ν 0 = ( π 2 κ 0 / 2 ) d p 0 d q 0 ,
K ( 2 , ψ 0 ) 1 4 0 ψ 0 d θ 0 θ 0 π / 2 π / 2 d υ 0 cos θ 0 .
K ( n , ψ 0 ) = 0 ψ 0 d θ 0 cos θ 0 [ π 2 ( θ 0 π 2 ) ] = 1 4 0 ψ 0 ( π θ 0 ) cos θ 0 d θ 0 = 1 4 [ ( π θ 0 ) sin θ 0 | 0 ψ 0 0 ψ 0 sin θ 0 ( d θ 0 ) ] .
K ( 2 , ψ 0 ) = ( 1 / 4 ) [ ( π ψ 0 ) sin ψ 0 + ( 1 cos ψ 0 ) ] .
F = ( π 2 4 ) n 1 κ 0 2 cos θ 0 ( i = 1 n 2 I ( B i + 1 , z > 0 ) C i ) V ( n 2 )
F N 1 N i = 1 N F ( i ) , F 2 N 1 N i = 1 N [ F ( i ) ] 2 ,
K ( n , ψ 0 ) F N ± a [ F 2 N F N 2 ] 1 / 2 N 1 / 2 ,
K ( 2 , ψ 0 ) ( π / 4 ) ψ 0 ( ψ 0 1 ) .
log K ( 2 , ψ 0 ) log ( π / 4 ) + log ψ 0 , ( ψ 0 1 )
y = a x b ,
log y = log a + b log x ,
K ( 2 , ψ 0 ) 0.7854 × ψ 0 ,
K ( 4 , ψ 0 ) 0.1673 × ψ 0 2 ,
K ( 5 , ψ 0 ) 0.0288 × ψ 0 2 ,
K ( 6 , ψ 0 ) 0.0483 × ψ 0 2 .
K ( 2 , ψ 0 ) 0.7854 × ψ 0 ,
K ( 4 , ψ 0 ) 0.1785 × ψ 0 2 ,
K ( 5 , ψ 0 ) 0.0335 × ψ 0 2 ,
K ( 6 , ψ 0 ) 0.0613 × ψ 0 2 .
K ( 3 , ψ 0 ) 0.33 × ψ 0 2 ln ψ 0 = 0.33 × ψ 0 2 ln ( 1 / ψ 0 ) ( 10 5 ψ 0 10 3 ) .
z * β s c t / 2 .
J n * ( z * ) J n ( t ) J n ( 2 z * / β s c ) ,
J n * ( z * ) / J 1 * ( z * ) = [ 2 n 1 K ( n , ψ 0 ) ] ( z * ) n 1 .
log [ J n * ( z * ) / J 1 * ( z * ) ] = log [ 2 n 1 K ( n , ψ 0 ) ] + ( n 1 ) log z * .

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